2016/05/19 by Rupam Barman, Barman, Rupam, Archit Pal Singh Sachdeva +1
Mathematics · #11P83 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11P83
paper · pdf · doi:10.48550/arxiv.1605.06037
First draft. Comments are welcome
arxiv created 2016/05/19 · arxiv updated 2016/05/20
Ramanujan's congruence p(5k+4) ≡ 0 \pmod 5 led Dyson \citedyson to conjecture the existence of a measure "rank" such that p(5k+4) partitions of 5k+4 could be divided into sub-classes with equal cardinality to give a direct proof of Ramanujan's congruence. The notion of rank was extended to rank differences by Atkin and Swinnerton-Dyer \citeatkin, who proved Dyson's conjecture. More recently, Mao proved several equalities and inequalities, leaving some as conjectures, for rank differences for partitions modulo 10 \citemao10 and for M2 rank differences for partitions with no repeated odd parts modulo 6 and 10 \citemaom2. Alwaise et. al. proved four of Mao's conjectured inequalities \citeswisher, while leaving three open. Here, we prove a limited version of one of the inequalities conjectured by Mao.